# Citizens and rebels: a twin prime related categorization of composites

Assuming Goldbach's conjecture and denoting by $$r_{0}(n)$$ the quantity $$\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$$ for a large enough composite integer $$n$$, consider the sequence $$(u_n)_n$$ such that $$u_0$$ is a given composite and $$u_{n+1}:=u_n.r_{0}(u_n)$$. If the sequence provably converges, we say $$u_0$$ is a "citizen", otherwise call it a "rebel". I investigated the considered sequences for $$u_0$$ below 121 and found some 25 rebels : I let the algorithm go on for 30 steps and made sure the last computed $$r_{0}$$ was not 1. This suggests there are more citizens that rebels, and actually a proof of the existence of infinitely many citizens would imply the twin prime conjecture.

My question is : are there solid arguments leading to a positive density of citizens among the composites? If yes can we estimate a lower bound thereof?